変数と関数いろいろな関数逆三角関数、正弦と余弦と正接と余接、その和

学習環境

微分積分演習〈理工系の数学入門コース/演習新装版〉 (和達三樹(著)、十河清(著)、岩波書店)の第2章(変数と関数)、2-1(いろいろな関数)、問題3の解答を求めてみる。

\begin{array}{l} - 1 < x < 1 \\ y_{1} = \arcsin x \\ y_{2} = \arccos x \end{array}

とおく。

\begin{array}{l} x = \sin y_{1} \\ x = \cos y_{2} \end{array}

このとき、

\sin (y_{1} + y_{2}) = \sin y_{1} \cos y_{2} + \cos y_{1} \sin y_{2}

= x^{2} + \sqrt{1 - \sin^{2} y_{1}} \sqrt{1 - \cos^{2} y_{2}}

= x^{2} + 1 - x^{2}

= 1

よって、

\sin (y_{1} + y_{2}) = 1

ゆえに、

\begin{array}{l} y_{1} + y_{2} = \frac{π}{2} \\ \arcsin x + \arccos x = \frac{π}{2} \end{array}

\begin{array}{l} y_{1} = \arctan x \\ y_{2} = arccot \frac{1}{x} \end{array}

とおくと、

\begin{array}{l} \tan y_{1} = x \\ \frac{1}{x} = \cot y_{2} = \frac{1}{\tan y_{2}} \end{array}

x = \tan y_{2}

よって、

\begin{array}{l} \tan y_{1} = \tan y_{2} \\ y_{1} = y_{2} \\ \arctan x = arccot \frac{1}{x} \end{array}

y = arccot x

とおくと

x = \cot y = \tan (\frac{π}{2} - y)

\arctan x = \frac{π}{2} - y

\arctan x = \frac{π}{2} - arccot x

\arctan x + arccot x = \frac{π}{2}

コード(Wolfram Language, Jupyter)

ArcSin[x] + ArcCos[x]

% == Pi / 2

Simplify[%]

Plot[ArcSin[x] + ArcCos[x], {x, -5, 5}]

ArcTan[x] == ArcCot[1/x]

Simplify[%]

Plot[ArcTan[x] - ArcCot[1/x], {x, -5, 5}]

ArcTan[x] + ArcCot[x]

Plot[%, {x, -5, 5}]

ArcTan[0] + ArcCot[0]